Answer:
(i)
f(x) = [x] on [5, 9].
Here
f(x) is a greatest integer function, which is discontinuous at integral points
5, 6, 7, 8 and 9 [5,9].
Since
f(x) does not satisfy all the conditions of Roll?s theorem therefore Rolle?s
theorem is not valid for the given function.
Although
f?(x) = 0 for non integral points does
not satisfies all the conditions of Rolle?s theorem.
Hence
converse of Roll?es theorem, is not true.
(ii)
f(x) = [x] on [?2, 2]
Here
f(x) is a greatest integer function which is not continuous at points ?2, ?1,
0, 1, 2
Since
f(x) does not satisfy all the conditions of Rolle?s theorem. Therefore.
Therefore Rolle?s theorem is not valid for the given function.
In
this case, the converse of Rolle?s therorem is not true.
(iii)
f(x) = x2 ? 1 on [1, 2]
f(1)
= 1 ? 1 = 0, f(2) = 4 ? 1 = 3
Since
f(1) theorem
f(x) does not satisfy all the conditions of Rolle?s theorem. Hence Rolle?s
theorem is not applicable for the given function.
Converse of
Rolle?s theorem is not true here.
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